ω = ω(k), and the group velocity can be written: In a medium such as classical vacuum the dispersion relation for electromagnetic waves is: where c0 is the speed of light in classical vacuum. For this case, the phase and group velocities both are c0. In so-called dispersive media the dispersion relation can be a complicated function of wavevector, and the phase and group velocities are not the same. For example, for several types of waves exhibited by atomic vibrations (phonons) in GaAs, the dispersion relations are shown in the figure for various directions of wavevector k. In the general case, the phase and group velocities may have different directions. In function approximation In condensed matter physics an energy eigenfunction for a mobile charge carrier in a crystal can be expressed as a Bloch wave: where n is the index for the band (for example, conduction or valence band) r is a spatial location, and k is a wavevector. The exponential is a sinusoidally varying function corresponding to a slowly varying envelope modulating the rapidly varying part of the wavefunction un,k describing the behavior of the wavefunction close to the cores of the atoms of the lattice. The envelope is restricted to k-values within a range limited by the Brillouin zone of the crystal, and that limits how rapidly it can vary with location r. In determining the behavior of the carriers using quantum mechanics, the envelope approximation usually is used in which the Schrödinger equation is simplified to refer only to the behavior of the envelope, and boundary conditions are applied to the envelope function directly, rather than to the complete wavefunction. For example, the wavefunction of a carrier trapped near an impurity is governed by an envelope function F that governs a superposition of Bloch functions: where the Fourier components of the envelope F(k) are found from the approximate Schrödinger equation. In some applications, the periodic part uk is replaced by its value near the band edge, say k=k0, and then: In diffraction patterns Diffraction patterns from multiple slits have envelopes determined by the single slit diffraction pattern. For a single slit the pattern is given by: where α is the diffraction angle, d is the slit width, and λ is the wavelength. For multiple slits, the pattern is where q is the number of slits, and g is the grating constant. The first factor, the single-slit result